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Mirrors > Home > MPE Home > Th. List > Mathboxes > oppcthin | Structured version Visualization version GIF version |
Description: The opposite category of a thin category is thin. (Contributed by Zhi Wang, 29-Sep-2024.) |
Ref | Expression |
---|---|
oppcthin.o | ⊢ 𝑂 = (oppCat‘𝐶) |
Ref | Expression |
---|---|
oppcthin | ⊢ (𝐶 ∈ ThinCat → 𝑂 ∈ ThinCat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppcthin.o | . . . 4 ⊢ 𝑂 = (oppCat‘𝐶) | |
2 | eqid 2740 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | 1, 2 | oppcbas 17779 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝑂) |
4 | 3 | a1i 11 | . 2 ⊢ (𝐶 ∈ ThinCat → (Base‘𝐶) = (Base‘𝑂)) |
5 | eqidd 2741 | . 2 ⊢ (𝐶 ∈ ThinCat → (Hom ‘𝑂) = (Hom ‘𝑂)) | |
6 | simpl 482 | . . . 4 ⊢ ((𝐶 ∈ ThinCat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ ThinCat) | |
7 | simprr 772 | . . . 4 ⊢ ((𝐶 ∈ ThinCat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶)) | |
8 | simprl 770 | . . . 4 ⊢ ((𝐶 ∈ ThinCat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶)) | |
9 | eqid 2740 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
10 | 6, 7, 8, 2, 9 | thincmo 48702 | . . 3 ⊢ ((𝐶 ∈ ThinCat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ∃*𝑓 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) |
11 | 9, 1 | oppchom 17776 | . . . . 5 ⊢ (𝑥(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑥) |
12 | 11 | eleq2i 2836 | . . . 4 ⊢ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ↔ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) |
13 | 12 | mobii 2551 | . . 3 ⊢ (∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) |
14 | 10, 13 | sylibr 234 | . 2 ⊢ ((𝐶 ∈ ThinCat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦)) |
15 | thincc 48697 | . . 3 ⊢ (𝐶 ∈ ThinCat → 𝐶 ∈ Cat) | |
16 | 1 | oppccat 17784 | . . 3 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) |
17 | 15, 16 | syl 17 | . 2 ⊢ (𝐶 ∈ ThinCat → 𝑂 ∈ Cat) |
18 | 4, 5, 14, 17 | isthincd 48710 | 1 ⊢ (𝐶 ∈ ThinCat → 𝑂 ∈ ThinCat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∃*wmo 2541 ‘cfv 6575 (class class class)co 7450 Basecbs 17260 Hom chom 17324 Catccat 17724 oppCatcoppc 17771 ThinCatcthinc 48692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-1st 8032 df-2nd 8033 df-tpos 8269 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-2 12358 df-3 12359 df-4 12360 df-5 12361 df-6 12362 df-7 12363 df-8 12364 df-9 12365 df-n0 12556 df-z 12642 df-dec 12761 df-sets 17213 df-slot 17231 df-ndx 17243 df-base 17261 df-hom 17337 df-cco 17338 df-cat 17728 df-cid 17729 df-oppc 17772 df-thinc 48693 |
This theorem is referenced by: (None) |
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